# [FOM] Infinitesimal calculus

joeshipman@aol.com joeshipman at aol.com
Tue May 26 20:24:34 EDT 2009

I disagree. Real numbers can be explained to high school students
perfectly soundly in terms of successive rational approximations
(decimal expansions, binary expansions, continued fractions, etc.) to
points on a geometric line. When this is done, they have a consistent
intuition for "what real numbers are", an intuition which makes a proof
of the least upper bound property quite easy to understand. There is no
philosophical difficulty at this level, unless you try to get across
the point that physical space might not have a structure that is
precisely captured by this definition.

On the other hand, when you use infinitesimals you have to be able
either to say "what an infinitesimal is" and distinguish infinitesimals
from the standard reals in a way that is intuitive and not merely
formal, or else you have to deny the Archimedean axiom and say that
infinitesimals are just as real as standard real numbers.  This CAN be
done in an intuitive way following Conway's "On Numbers and Games", but
all the attempts I have seen to use infinitesimals to develop Calculus
for students do not do this, instead  treating them purely formally in
a system where you can't get your hands on an infinitesimal directly.
This is unsatisfactory for high-school level students who aren't
usually comfortable with introducing elements that can be regarded as
fictitious.

(In Conway's system you just extend Dedekind cuts into the transfinite,
so each individual number can be expressed as a map from some ordinal
to the set {+, -} and when the ordinal is at most \omega you recover
the classical real numbers as a substructure.)

-- JS

-----Original Message-----
From: David Ross <ross at math.hawaii.edu>

I don't really want to engage yet again in the argument as to whether
it is
better or not to teach calculus with infinitesimals, I just want to
point
out that some of the remaining arguments against doing so do not hold
up to
strong scrutiny.  When we made the switch in the US 100 years ago, it
was
for reasons of rigor; now, however, it is ultimately just a matter of
taste.